Puzzle for December 18, 2018 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* AB, BC, and CD are 2-digit numbers (not A×B, B×C, or C×D).
Scratchpad
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Hint #1
eq.6 may be written as: B + C + D = 10×C + D Subtract both C and D from each side of the equation above: B + C + D - C - D = 10×C + D - C - D which simplifies to B = 9×C
Hint #2
In eq.3, substitute B - F for D (from eq.2): A + B = B - F + E + F which becomes A + B = B + E Subtract B from each side of the above equation: A + B - B = B + E - B which makes A = E
Hint #3
eq.5 may be written as: 10×B + C = 10×A + B + C - D - F Subtract both B and C from each side of the above equation: 10×B + C - B - C = 10×A + B + C - D - F - B - C which becomes eq.5a) 9×B = 10×A - D - F
Hint #4
From eq.2, substitute (B - F) for D in eq.5a: 9×B = 10×A - (B - F) - F which is equivalent to 9×B = 10×A - B + F - F which becomes 9×B = 10×A - B Add B to each side: 9×B + B = 10×A - B + B which simplifies to 10×B = 10×A Divide both sides by 10: 10×B ÷ 10 = 10×A ÷ 10 which makes B = A making 9×C = B = A = E
Hint #5
Substitute 9×C for A in eq.4: 9×C - C = F + C Subtract C from both sides: 9×C - C - C = F + C - C which makes 7×C = F
Hint #6
Substitute 9×C for B, and 7×C for F in eq.2: 9×C - 7×C = D which makes 2×C = D
Solution
Substitute 9×C for A and B and E, 2×C for D, and 7×C for F in eq.1: 9×C + 9×C + C + 2×C + 9×C + 7×C = 37 which simplifies to 37×C = 37 Divide both sides by 37: 37×C ÷ 37 = 37 ÷ 37 which means C = 1 making A = B = E = 9×C = 9 × 1 = 9 D = 2×C = 2 × 1 = 2 F = 7×C = 7 × 1 = 7 and ABCDEF = 991297